Try this then:
$('#yourButton').click(function(){
$('#search-input').keyup();
});
When you click your button, it will trigger the keyup event on your input.
The library you are using is listening to keyup event:
function registerInput(){
opt.searchInput.addEventListener('keyup', function(e){
if( e.target.value.length == 0 ){
emptyResultsContainer()
return
}
render( searcher.search(store, e.target.value) )
})
}
Found the code from Github source.
Another way to change the library. Modify src/index.js file like this:
$('#yourbutton').click(function(){
render( searcher.search(store, opt.searchInput.value) );
})